the mathematics of musical instruments

The Mathematics of Musical InstrumentsRachel W. Hall and Kresimir Josic

1. INTRODUCTION The history of musical instruments goes back tens of thou-sands of years. Fragments of bone flutes and whistles have been found at Neanderthalsites. Recently, a 9,000-year-old flute found in China was shown to be the world’soldest playable instrument (pictures and a recording of this flute are available athttp://www.bnl.gov/bnlweb/flutes.html). These early instruments show that peoplehave long been concerned with producing pitched sound—that is, sound containingpredominantly a single frequency. Indeed, finger holes on the flutes indicate that theseprehistoric musicians had some concept of a musical scale.

The study of the mathematics of musical instruments dates back at least to thePythagoreans, who discovered that certain combinations of pitches that they consid-ered pleasing corresponded to simple ratios of frequencies such as 2:1 and 3:2. Theproblems of tuning, temperament, and acoustics have since occupied some the bright-est minds in the natural sciences. Marin Marsenne’s treatise on tuning and acous-tics Harmonie Universelle (1636) [19], H. v. Helmholtz’s On the Sensations of Tone(1870) [15], and Lord Rayleigh’s seminal The Theory of Sound (1877) [21] are justthree outstanding examples.

Many pages have been written on this subject. We mean to present an overview andlet the interested reader find more detailed discussions in the references, and on ourweb site www.sju.edu/∼rhall/newton.

Figure 1. Musicologist Ola Kai Ledang playing the willow flute

2. THE WILLOW FLUTE In this section we consider the physical properties ofa Norwegian folk flute called the seljefløyte, or willow flute. This instrument can beconsidered “primitive” in that it does not rely on finger holes to produce differentpitches. Rather, by varying the strength with which he or she blows into the flute, theplayer selects from a series of pitches called harmonics, whose frequencies are integermultiples of the flute’s lowest tone, called the fundamental. The willow flute’s scale isapproximately a major scale with a sharp fourth and flat sixth, and plus a flat seventh.

The willow flute is a member of the recorder family, though it is held transversally.The flute is constructed from a hollow willow branch (or, more recently, a PVC pipe;see the quirky but informative web site http://www.geocities.com/SoHo/Museum/-

April 2001] 347

4915/SALLOW.HTM for instructions). One end is open and the other contains a slotinto which the player blows, forcing air across a notch in the body of the flute. Theresulting vibration creates standing waves inside the instrument whose frequency de-termines the pitch. The recorder has finger holes that allow the player to change thefrequency of the standing waves, but the willow flute has no finger holes. However, itis evident from the tune Willow Dance (Figure 2), as performed by Hans Brimi on thewillow flute [8], that quite a number of different tones can be produced on the willowflute. How is this possible?

Figure 2. Transcription of Willow Dance, as performed by Hans Brimi

The answer lies in the mathematics of sound waves. Let u be the pressure in thetube, let x be the position along the length of the tube, and let t be time. Since thepressure across the tube is close to constant, we can neglect that direction. We chooseunits such that the pressure outside the tube is 0.

The one-dimensional wave equation

a2 ∂2u

∂x2= ∂2u

∂t2

provides a good model of the behavior of air molecules in the tube; a is a positiveconstant. Since both ends of the tube are open, the pressure at the ends is the same asthe outside pressure. That is, if L is the length of the tube, u(0, t) = 0 and u(L , t) = 0.Solutions to the wave equation are linear combinations of solutions of the form

u(x, t) = sinnπx

L

(b sin

anπ t

L+ c cos

anπ t

L

)

where n = 1, 2, 3, . . . , and b and c are constants. The derivation of this solution maybe found in most textbooks on differential equations; see [7].

How does our solution predict the possible frequencies of tones produced by theflute? For now, let’s just consider solutions that contain one value of n. Fix n and xand vary t . The pressure varies periodically with period 2L/an. Therefore,

frequency = an

2L

for n = 1, 2, 3, . . . .

This formula suggests that there are two ways to play a wind instrument: eitherchange the length L , or change n (for stringed instruments, which are also governedby the one-dimensional wave equation, there are three ways, since one can also changethe value of a by changing the tension on the string—for example, by string bending—or by substituting a string of different density). Varying L continuously, as in the slide

348 c© THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

trombone or slide whistle, produces continuous changes in pitch. The more commonway to change L is to make holes in the tube, which allow for discrete changes in pitch.The other way to vary the pitch is to change n—that is, to jump between solutions ofthe wave equation. The discrete set of pitches produced by varying n are the harmonics.Specifically, the pitch with frequency an/2L is called the nth harmonic; if n = 1 thepitch is the fundamental or first harmonic. The term overtone is also used to describethese pitches, but the nth harmonic is called the (n − 1)st overtone.

The sequence of ratios of the frequency of the fundamental to the successive har-monics is 1:1, 1:2, 1:3, 1:4, . . . (note the connection to the harmonic series 1 + 1

2 +13 + 1

4 + · · ·). If the first harmonic is a C, then the next five harmonics are C′, G′, C′′,E′′, and G′′, where each prime denotes the pitch one octave higher. The fourth, fifth,and sixth harmonics form what is called a major chord, one of the primary buildingblocks of Western music. However, this solution still doesn’t completely explain thewillow flute.

Let’s take a closer look at the willow flute player’s right hand (Figure 1). The posi-tion of his fingers allows him to cover or uncover the hole at the end of the flute: he canchange the boundary condition at that end. At the closed end, air pressure is constantin the x direction, so the boundary conditions become u(0, t) = 0 and ux(L , t) = 0.Solving the wave equation as before, we get a set of solutions for which

frequency = an

4L

for n = 1, 3, 5, 7, . . .. Since the original value of frequency was an/2L, closing theend has dropped the fundamental an octave and restricted the harmonics to odd mul-tiples of the fundamental frequency. Combining the harmonics produced with endclosed and with end open, we see that in the third octave (relative to the fundamentalof the open pipe) there is a nine-note scale available, which we call the flute’s playingscale. As an aid to visualization, these two sets of harmonics, along with the ratios oftheir frequencies to the fundamental of the open pipe, are shown in their approximatepositions on a piano keyboard in Figure 3 (assuming the fundamental of the open pipeis tuned to a C). However, it should be emphasized that pianos are not commonly tunedto the willow flute’s scale!

2:11 1:72:7

2:32:1 1:1 1:2 2:5 1:3 1:4 1:5 1:6 1:82:9 2:15

2:13

==

Pitches produced with end openPitches produced with end closed

Figure 3. Approximate location of the willow flute’s pitches on a piano

How necessary is this second set of harmonics? The harmonics produced by theopen pipe in the fourth octave from its fundamental form the same scale as the com-bined open- and closed-end scale, but an octave higher. In fact, if we don’t care whichoctave we’re in, the willow flute can theoretically produce pitches arbitrarily close toany degree of the scale, even without changing the boundary condition. However, thissolution is impractical. The higher harmonics are not only less pleasing to the ear, but

April 2001] 349

also more difficult to control. In order to produce the rapid note changes required forthe Willow Dance, the player needs the second set of harmonics. Some willow fluteplayers extend this technique by covering the end hole only halfway to produce anintermediate set of pitches, or by continuously changing the boundary condition toproduce a continuous change in pitch.

So far, we have considered only those solutions to the wave equation of the form

u(x, t) = sinnπx

L

(b sin

anπ t

L+ c cos

anπ t

L

)

A more general solution is a linear combination of several of these. The terms contain-ing the smallest values of n generally have the greatest amplitude (scalar coefficient)and determine the pitch and character of the sound. In particular, the fundamentalpredominates and it is perceived as the pitch of the sound. The relative volumes ofthe harmonics help us to distinguish the sounds of different musical instruments. Forexample, the clarinet’s sound contains only odd harmonics, as does the sound of thewillow flute with the end closed. Sethares’ fascinating book [25] proposes that West-ern music uses scales based on small integer ratios of frequencies precisely becausethe sound of winds and strings consists of harmonics. When two such instrumentsplay notes from the same scale, many of the harmonics produced by the instrumentscorrespond, creating an effect pleasing to the ear.

To describe the acoustic properties of instruments fully, it is also necessary to takeinto account nonlinear effects. This is still a very active area of research, and a goodoverview may be found in [12].

3. FROM MELODY TO HARMONY: KEYBOARD INSTRUMENTS In thissection, we use the willow flute as the jumping-off point for a discussion of scaleconstruction. The willow flute’s playing scale is appealing mathematically becauseeach ratio of frequencies within the scale can be expressed as a ratio of small integers.And, since the music traditionally played on the willow flute is exclusively melodicand centered on the key of its fundamental, we are less concerned with how the noteswithin its scale are related to one another. However, when we try to use this systemto design keyboard instruments, problems arise. For example, we would like the 4:5:6relationship of the major chord to be replicated in several locations on our keyboard,and we would like our instrument to sound “good” in several different keys. It turnsout that these goals are not simultaneously achievable. The story of the attempts toresolve this issue illustrates one of the most interesting intersections of mathematicsand aesthetics.

Just intonation We begin by writing down the ratios of frequencies of the nine notesin the willow flute’s playing scale to the first note of that scale (Figure 4). These arecomputed by finding the relationship of each note to the flute’s first harmonic and thendividing to find their relationship to each other. Observe that the sequence of ratios inthis scale can be written 8:8, 9:8, 10:8, . . . .

A major chord is comprised of three notes whose ratio of frequencies (give or takean octave) is 4:5:6. We see that there are two major chords in the willow flute’s scale:the chord formed by the first, third, and fifth notes in its scale (called the I chord),and the chord formed by the fifth, eighth, and second notes (called the V chord). Justintonation is based on an eight-note scale that may be decomposed into three majorchords: I, V, and IV, which contains the fourth, sixth, and eighth notes of the justintonation scale. Many versions of just intonation were proposed between the 15thand the 18th century, most differing on how to construct the remaining notes in the


2:1

15:8

5:3

3:2

4:3

5:4

9:8

1:1 1:1

9:8

5:4

11:83:213:8

15:8

2:1

7:4

just willowintonation flute

Figure 4. Comparison of just intonation and the willow flute’s playing scale

chromatic scale [3], [5]. A history of the various systems and practical guidelines forimplementing them, as well as discussion of Mersenne’s work on this problem, isfound in [16].

Just intonation has several problems. One of the most glaring is the ratio of thesixth to second degrees of the scale, which is 40:27, rather than 3:2. When just into-nation is used, the same note may have a different pitch in several keys. For instance,the ratio of A to G is equal to 10:9 when we’re playing in C, rather than 9:8 whenwe’re playing in G. The players of stringed and wind instruments can make thesesmall adjustments in pitch as they play; however, a different system must be devisedfor instruments with fixed pitch. Various compromises have been proposed, includingtempered scales, which involve adjustments to just intonation. Another solution forkeyboard instruments is to add keys, or to allow the pitch to be altered by applyinglevers or pedals. Many ingenious methods were developed to translate these ideas intopractice. The earliest is the organ of St. Martin’s at Lucca, which has separate keys forE� and D. The “Enharmonium” of Tanaka separated the octave into 312 notes [3].One of the few instruments of this type in use today is the English concertina, whichhas separate buttons for E� and D and for A� and G. Most modern concertina playersopt to have their instruments tuned to equal temperament, however.

The Pythagorean scale In the previous section we saw that intervals are formed bymultiplying a fundamental frequency by a rational number. Pythagoras discovered thatthe 2:1 ratio of an octave and the 3:2 ratio of a fifth are particularly consonant and usedthem as the basis for a scale. His construction avoids the problem of some fifths beingout of tune in just intonation. The idea is to start with a fundamental frequency andmultiply repeatedly by 3

2 to obtain other notes in the scale. Two notes that are an octaveapart represent the same degree of the scale. Therefore, if multiplying a frequency fby 3

2 gives us a frequency that is not in the octave in which we started—that is, if32 × f > 2—we can divide the result by 2 to return to the original octave.

It is convenient to work with logarithms of base 2 of a given frequency, rather thanthe frequency itself. If we choose units such that middle C has frequency 1, then in thelogarithmic units middle C has logarithmic frequency log2 1 = 0, while C′, an octaveabove middle C, has logarithmic frequency 1 since 21 = 2. Setting x = log2 f and

April 2001] 351

taking the logarithm base 2 we obtain

x → x + log2

3

2

as the mapping taking a tone to its fifth in logarithmic units. Since dividing by 2 cor-responds to subtracting 1 on the logarithmic scale, and since we subtract 1 only ifx + log2

32 > 1, we obtain the following map on the interval [0, 1]:

x → x + log2

3

2(mod 1).

By identifying the endpoints of the interval [0, 1], this map can be thought of asan irrational rotation of a circle. It is known that an initial point x0 never returns toitself under the iteration of such a map. Rather, its images fill the circle densely [11].Therefore, if we move by fifths, we never return to the the frequency with which westarted. This fact has unfortunate consequences for the construction of a scale, as wasdiscovered by the Pythagoreans. This problem is most apparent in instruments withfixed pitch.

Equal temperament Equal temperament involves approximating an irrational rota-tion of the circle by a rational one. There is a natural geometric way to think about thisapproximation. The graph of the line y = µx intersects the vertical lines x = q, whereq is a positive integer, at the points µq. The decimal part of this number is exactly theq-th iterate of 0 under the rotation map x → x + µ (mod 1).

If µ is irrational this line does not pass through any points of the lattice Z × Z,and therefore an irrational rotation of the circle has no periodic orbits. A line passingthrough a point (q, p) ∈ Z × Z that lies close to y = µx gives rise to a rotation x →x + p/q (mod 1), which approximates the rotation x → x + µ (mod 1). If thefraction p/q in the approximate mapping is in reduced form, the orbit of any pointhas period q, and the points of the orbit are distributed uniformly around the circle.Thus, we can use this approximation to divide the octave into q equal parts. Scalesconstructed in this way are called equally tempered.

The following is a geometric way to find a sequence of points (q, p) ∈ Z × Zthat are successively closer to y = µx . This discussion follows F. Klein’s construc-tion in [1]. Imagine a string attached to infinity extending to the origin along the liney = µx . Also imagine that a nail is driven through each point with positive integercoordinates in the plane. If we pull the free end of the string up or down, it touches thenails that are closest to the line y = µx . The region bounded by the pulled-up stringand the line x = 1 is the convex hull of the points above y = µx . The pulled-downstring bounds the convex hull of the points below y = µx .

For example, if µ = log232 , the string touches (1, 1), (5, 3), (41, 24), (306, 179), . . .

when it is pulled up, and (2, 1), (12, 7), (53, 31), . . . when it is pulled down, as shownin Figure 5, where the string bounds the gray areas. Therefore log2

32 is approximated

well by the sequence 1, 12 , 3

5 , 712 , 24

41 , 3153 , 179

306 , . . . .

The meaning of “approximated well” can be made precise. Consider the followingconstruction: Let e−1 = (0, 1) and e0 = (1, 0). If ek−1 and ek are given, let ek+1 bethe vector obtained by adding ek to ek−1 as many times as possible without crossingy = µx .

Proposition 1. The oriented area of the parallelogram spanned by the vectors ek−1

and ek is (−1)k , when orientation is taken into account.


e0 2 4 6 8 10 12

2

4

6e4

e1e−1

e2

upperconvex hull

lower

convex hull

e3

Figure 5. Approximation of the line y = (log232 )x

Proof. Every subsequent parallelogram shares a side and altitude with its predecessor.

Corollary 1. The points ek, k > 0, are extreme points of either the upper or lowerconvex hulls.

Proof. If the points ek−1 and ek were not on the convex hull, the parallelogram formedby the vectors ek−1 and ek would have to contain a point in Z × Z. By Pick’s Theo-rem [26] such a parallelogram has area greater than 1, contradicting Proposition 1.

The following is another straightforward corollary [1].

Corollary 2. If qk and pk are the coordinates of ek, k > 0, then∣∣∣∣µ − pk

qk

∣∣∣∣ <1

q2k

.

This corollary shows that the numbers we obtained in the geometric constructionsare the convergents obtained in the continued fraction expansion of µ, and in this sensethey are the best rational approximations to µ. Detailed discussions of continued frac-tions can be found in [14], while their applications to music are discussed in [2]. Inparticular, note that log2

32 is a transcendental number, which means that the complex-

ity of its rational approximations increases rapidly.There are several things to consider when choosing any of these approximations as

the basis for a scale. The period of a rational rotation is determined by the denominatorof the fraction p/q. A large denominator leads to a scale with many notes. This is im-practical, due to the physical constraints of instruments and our inability to distinguishtones that are very close in pitch.

It is at least in part due to these considerations that the approximation log223 ≈ 7

12is used as the basis of Western music. Figure 6 shows how the evenly spaced tonesobtained from this approximation divide the octave into 12 and 41 parts (41 parts pro-viding the next best approximation). The gray dots represent just intonation. It is evi-dent that the just intonation scale is fairly well approximated by the equally temperedtwelve-note scale. This is somewhat fortuitous, as we have made no effort to approxi-mate the 5:4 ratio of a third in our construction. Whether the benefits obtained by this

April 2001] 353

Figure 6. The circle divided into 12 and 41 equal parts

construction outweigh the price that we have to pay in having all intervals “impure” isstill a subject of debate. A Mathematica application that explores this construction inmore detail is available at www.sju.edu/∼rhall/newton.

There are other ways to construct equally tempered scales. We could try to find ra-tional numbers with equal denominators that approximate both the fifth and third well.The associated rotation would divide the circle into equal parts. This approach leads tothe theory of higher-dimensional continued fractions, which is still a subject of muchresearch; see [2] or [18]. Many equally-tempered scales were explored in the past,from the 17-note Arabian scale to the 87-note division praised by Bosanquet. EasleyBlackwood [6] has written compositions for each of the equally tempered scales con-taining 13 to 24 tones. J. M. Barbour [3] and D. Benson [5] present excellent historicalreviews of this subject. The reader is invited to compare the merits of the differentsubdivisions using the Mathematica program available on the authors’ web page.

The scale of twelve equally tempered notes leads to another interesting question.The distance between the frets of an equally tempered stringed instrument such as aguitar or a lute has to be scaled by the ratio 2

112 : 1. Since 2

112 = (2

13 )

14 this problem

is equivalent to duplicating a cube, a task that cannot be accomplished by Euclideanmethods. Constructing this ratio with the methods of measurement available in the 16thand 17th century was a difficult task and finding an approximation was of considerableutility. Several interesting approaches, including ingenious constructions by GalileoGalilei’s father and Stahle are discussed in [4].

4. DRUMS AND OTHER HIGHER-DIMENSIONAL INSTRUMENTS So farwe have considered only instruments that are essentially one-dimensional. All stringedand wind instruments fall into this category. Percussion instruments such as drums andbells do not. Why is that? Let us think of a drum with a circular drumhead as a circulardomain of radius R around the origin in R2 that obeys the wave equation with fixedboundary. Using polar coordinates (r, φ) and separation of variables it can be shownthat the transversal displacement of the drumhead at time t is given by F(r, φ, t) =g(t) f1(r) f2(φ) where

g′′(t) + c2λg(t) = 0 f ′′2 (φ) + µ f2(φ) = 0 (1)

f ′′1 (r) + 1

rf ′1(r) +

(λ − µ

r 2

)f2(r) = 0. (2)


The constant c is related to the physical properties of the material, and λ and µ aredetermined from the conditions f2(−π) = f2(π) and f1(R) = 0. See [20] for moredetails.

The equations for g and f2 are easy to solve. The constraints on f2 force µ = m2,which means that (2) is exactly the m-th Bessel equation, whose solutions are given interms of the m-th Bessel function as f1(r) = Jm(r

√λ). Since the drumhead is fixed

along its boundary this means that f1(R) = Jm(R√

λ) = 0 and so λ can assume onlythe values

λn =(

x (m)n

R

)2

, (3)

where x (m)n are the zeros of the m-th Bessel equation. The different values of λ deter-

mine the frequencies of oscillation of the different modes, as in the case of the willowflute. Since the zeros are irrationally related, it follows that the frequencies of oscilla-tions of the drumhead cannot be rational multiples of each other. This is why drumsusing a freely oscillating circular membrane and one-dimensional instruments producenotes whose tonal character is discernably different.

One instrument that does not fit within this picture is the timpanum. The mem-brane of the timpanum is truly two-dimensional, and yet its sound is similar to that ofone-dimensional instruments. Unlike a tambourine, the timpanum has a closed bottomand its vibrations change the pressure in the cavity beneath the oscillating membrane.Therefore its membrane is not freely oscillating and additional nonlinear forcing termshave to be added to the wave equation to describe its behavior accurately. By carefullytuning the bowl beneath the membrane, the frequencies of the first few modes of vi-bration can be related as 2:3:4:5 [10].

Of course, we do not have to restrict ourselves to circular drums. We can con-sider the wave equation on a general domain D in R2 and look for solutions thatsatisfy F(x, y, t) = 0 on the boundary ∂ D. Separating variables as F(x, y, t) =�(t) �(x, y) lets one conclude that the general solution is of the form F(x, y, t) =sin(

√λt) �(x, y), where

∇2� + λ� = 0 in D and � = 0 on ∂ D. (4)

As we have seen before, a solution to this problem exists only for certain values of λ

known as eigenvalues. These eigenvalues depend on the shape of the drum D, and arethe squares of the frequencies of vibrations of the different modes.

In his beautiful article Can one hear the shape of a drum? M. Kac asked whethertwo drums with the same eigenvalues necessarily have the same shape [17]. Kacproved that certain characteristics of the domain, such as its area and circumference,are indeed determined by the eigenvalues. The general problem remained unsolved for24 years until Gordon et al. showed that two non-congruent drums can have the sameeigenvalues [13]. For an explicit construction of two such drums see [9].

We still have one dimension remaining: can we characterize the sound of three-dimensional instruments? All three-dimensional instruments fall into the class ofpercussion instruments. If we write the wave equation for some simple geometricshapes—for example, a rod—we can conclude that the frequencies of the differentmodes of vibration are not rationally related. Instruments of this type are less com-mon in Western music. Sethares [25] shows that the scales used by the Indonesiangamelan are related to the gamelan instruments’ spectra, which consist of tones that

April 2001] 355

are not rationally related to the fundamental. There are, however, three-dimensionalinstruments whose sounds are similar to the sounds of one-dimensional instruments:the marimba, the glockenspiel, claves, and others. There are several ways to achievethis. Some three-dimensional objects, such as rods, vibrate predominantly in a one-dimensional fashion. On the other hand, the bars on a marimba corresponding to lowertones have deep arches on one side. These are cut in such a way that the first twomodes of vibration of the bar are rationally related. Since the higher notes are abovethe 2000 Hz range, they are not as important in determining the perceived sound ofthose bars. Excellent descriptions of percussion instruments can be found in the worksof Rossing [22], [23], [24] (a popular treatment), and [12] (with Fletcher).

ACKNOWLEDGMENTS We thank Paul Klingsberg and Bill Sethares for their helpful comments, Ola KaiLedang for the photo, and David Loberg Code for information about the willow flute.

REFERENCES

1. V. I. Arnol’d, Geometrical methods in the theory of ordinary differential equations, 2nd ed., Springer-Verlag, New York, 1988. Translated from the Russian by Jozsef M. Szucs.

2. J. M. Barbour, Music and ternary continued fractions, Amer. Math. Monthly 55 (1948) 545–555.3. J. M. Barbour, Tuning and Temperament, Michigan State College Press, East Lansing, 1953.4. J. M. Barbour, A geometrical approximation to the roots of numbers, Amer. Math. Monthly 64 (1957)

1–9.5. D. Benson, Mathematics and Music, book in progress, available at

ftp://byrd.math.uga.edu/pub/html/index.html, 2000.6. E. Blackwood, Microtonal Etudes, Cedille Records, CDR 90000 018.7. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 4th

ed., John Wiley & Sons, Inc., New York, 1986.8. H. Brimi, “Willow Dance, 1994,” in The Sweet Sunny North, Shanachie Records 64057.9. S. J. Chapman, Drums that sound the same, Amer. Math. Monthly 102 (1955) 124–138.

10. R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, Effect of AirLoading on Timpani Membrane Vibrations, J. Acoust. Soc. Am. 76 (1984) 1336–1345.

11. R. L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Publishing Com-pany Advanced Book Program, Redwood City, CA, 1989.

12. N. H. Fletcher and T. D. Rossing, The physics of musical instruments, 2nd ed., Springer-Verlag, NewYork, 1998.

13. C. Gordon, D. L. Webb, and S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27(1992) 134–138.

14. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford UniversityPress, Oxford, 1980.

15. H. v. Helmholtz, On the sensations of tone as a physiological basis for the theory of music, Dover Publi-cations, New York, 1954. Originally published 1870, English translation by A. J. Ellis, 1875.

16. O. Jorgensen, Tuning, Michigan State University Press, East Lansing, 1991.17. M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966) 1–23.18. J. T. Kent, Ternary continued fractions and the evenly-tempered musical scale, CWI Newsletter 13 (1986)

21–33.19. M. Mersenne, Harmonie universelle, contenant la theorie et la pratique de la musique, Centre national

de la recherche scientifique, Paris, facsimile edition, 1963. Originally published 1636.20. M. A. Pinsky, Partial differential equations and boundary value problems with applications, 2nd ed.,

McGraw-Hill Inc., New York, 1991.21. J. W. S. Rayleigh, The theory of sound, 2nd ed., Dover Books, New York, 1945. Originally published

1877.22. T. D. Rossing, Acoustics of Percussion Instruments—part i, Phys. Teach. 14 (1976) 546–556.23. T. D. Rossing, Acoustics of Percussion Instruments—part ii, Phys. Teach. 15 (1977) 278–288.24. T. D. Rossing, The science of sound, 2nd ed., Addison-Wesley, New York, 1990.25. W. A. Sethares, Tuning, timbre, spectrum, scale, Springer-Verlag, New York, 1999.26. D. E. Varberg, Pick’s theorem revisited, Amer. Math. Monthly 92 (1985) 584–587.


RACHEL W. HALL is an assistant professor of mathematics at St. Joseph’s University in Philadelphia. Shereceived a B.A. in ancient Greek from Haverford College in 1991 and a Ph.D. in mathematics from the Penn-sylvania State University in 1999. Her field of research is operator algebras. She is also a folk musician andplays English concertina and piano with the trio Simple Gifts. Their award-winning 1999 album Time andAgain has received international airplay. In 1991, Rachel received a Watson Fellowship to study traditionaldance music in Norway, which is where she first encountered the willow flute.St. Joseph’s University, 5600 City Ave., Philadelphia, PA [email protected]

KRESIMIR JOSIC is a visiting assistant professor of mathematics at Boston University. He received a B.Sc.in physics and mathematics from the University of Texas at Austin in 1994 and a Ph.D. in mathematics fromthe Pennsylvania State University in 1999. His main research interest is applications of the theory of dynamicalsystems. He is also a jazz bass player.Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA [email protected]

limn→∞

n√

n = 1

By the arithmetic-geometric inequality, we have

0 ≤ n√

n − 1 = n − 1

nn−1

n + nn−2

n + · · · + 1

≤ n − 1

nn√

n1n + 2

n ···+ n−1n

= n − 1

n3n−1

2n

≤ 13√

n.

Contributed by J. Rooin, Institute for Advanced Studies in Basic Sciences,Gava Zang, Zanjan, Iran. [email protected]

April 2001] 357

the mathematics of musical instruments

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